derivative (x^3)/(4-x^2)

解

導関数

\frac{x ^{3}}{4 - x ^{2}}

\frac{12 x ^{2} - x ^{4}}{( 4 - x ^{2} ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x ^{3}}{4 - x ^{2}})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( x ^{3} ) ( 4 - x ^{2} ) - \frac{d}{d x} ( 4 - x ^{2} ) x ^{3}}{( 4 - x ^{2} ) ^{2}}

\frac{d}{d x} (x^{3}) = 3 x^{2}

\frac{d}{d x} (4 - x^{2}) = - 2 x

= \frac{3 x ^{2} ( 4 - x ^{2} ) - ( - 2 x ) x ^{3}}{( 4 - x ^{2} ) ^{2}}

簡素化

\frac{3 x ^{2} ( 4 - x ^{2} ) - ( - 2 x ) x ^{3}}{( 4 - x ^{2} ) ^{2}} : \frac{12 x ^{2} - x ^{4}}{( 4 - x ^{2} ) ^{2}}

= \frac{12 x ^{2} - x ^{4}}{( 4 - x ^{2} ) ^{2}}